Question 313: Gravitation

Question

Earth has mass ' $M_1$ ' radius ' $R_1$ ' and for moon mass ' $M_2$ ' and radius ' $R_2$ '. Distance between their centres is ' $r$ '. A body of mass ' $M$ ' is placed on the line joining them at a distance $\frac{\mathrm{r}}{3}$ from the centre of the earth. To project a mass ' $M$ ' to escape to infinity, the minimum speed required is

Accepted Answer

The binding energy of the body is given by
$$\begin{aligned}
	ext { B.E. } & =\frac{G M_1 M}{\frac{r}{3}}+\frac{\mathrm{GM}_2 \mathrm{M}}{\frac{2 r}{3}}=\frac{3 G M_1 M}{r}+\frac{3 \mathrm{GM}_2 M}{2 \mathrm{r}} \
& =\frac{3 G M}{r}\left[M_1+\frac{M_2}{2}ight]
\end{aligned}$$
If v is the velocity given to the body, then
$$\begin{aligned}
& \frac{1}{2} M v^2=\frac{3 G M}{r}\left[M_1+\frac{M_2}{2}ight] \
	herefore \quad & v=\left[\frac{6 G}{r}\left(M_1+\frac{M_2}{2}ight)ight]^{\frac{1}{2}}
\end{aligned}$$

Answer

$\left[\frac{2 G}{r}\left(M_2+\frac{M_1}{2}\right)\right]^{1 / 2}$

Answer

$\left[\frac{4 \mathrm{G}}{\mathrm{r}}\left(\mathrm{M}_1+\frac{\mathrm{M}_2}{2}\right)\right]^{1 / 2}$

Answer

$\left[\frac{3 G}{r}\left(M_1+M_2\right)\right]^{1 / 2}$

Answer

$\left[\frac{6 G}{r}\left(M_1+\frac{M_2}{2}\right)\right]^{1 / 2}$

Gravitation

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